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Properties

Label 1-1665-1665.544-r0-0-0
Degree $1$
Conductor $1665$
Sign $0.851 + 0.523i$
Analytic cond. $7.73222$
Root an. cond. $7.73222$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s − 7-s − 8-s + (−0.5 + 0.866i)11-s + (0.5 − 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s − 22-s + (0.5 + 0.866i)23-s + 26-s + (0.5 − 0.866i)28-s + (−0.5 + 0.866i)29-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s − 7-s − 8-s + (−0.5 + 0.866i)11-s + (0.5 − 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s − 22-s + (0.5 + 0.866i)23-s + 26-s + (0.5 − 0.866i)28-s + (−0.5 + 0.866i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1665 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.851 + 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1665 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.851 + 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1665\)    =    \(3^{2} \cdot 5 \cdot 37\)
Sign: $0.851 + 0.523i$
Analytic conductor: \(7.73222\)
Root analytic conductor: \(7.73222\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1665} (544, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1665,\ (0:\ ),\ 0.851 + 0.523i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.286549600 + 0.3637133036i\)
\(L(\frac12)\) \(\approx\) \(1.286549600 + 0.3637133036i\)
\(L(1)\) \(\approx\) \(0.9797787953 + 0.4409392203i\)
\(L(1)\) \(\approx\) \(0.9797787953 + 0.4409392203i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 - T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + T \)
61 \( 1 + T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 - T \)
79 \( 1 + T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.51538136846960901553428285288, −19.38682017375882774341320402389, −18.9500312700090269351621014168, −18.626858387186279359400442841809, −17.385505157531904029785963044976, −16.45432061115272498994595940821, −15.9106777660915392621313298279, −14.86200909192910238453269088226, −14.20027679771587411953904710836, −13.39855042715568157173489775940, −12.78844930417065421089947830388, −12.17397960333473293546300082030, −11.19754717125620910026204371878, −10.53971988836328339305239674111, −9.910333851863562780831256299549, −8.940186183623059690004779066920, −8.37398442689099000064209031924, −6.98463601570855912139636871490, −6.02369650524037862253746993725, −5.68287432515049145275408559348, −4.32891615213646590178986819346, −3.70546512535366673374954221993, −2.9312380210930196007650415550, −1.98259673768385658904590940690, −0.87182979165095141235418030109, 0.52733416805431184115790325309, 2.353500228774286127364171113482, 3.22400639091542928441067709748, 3.91664166640870818238561949919, 5.1333892606065074168132873159, 5.51855913149507343490986929292, 6.651151349666941861609473186324, 7.18853326177152536883585473971, 7.93939608467609362936498146964, 8.991133725634854465554465739487, 9.58396720007506114603782465401, 10.50166356533954262995933241723, 11.59929400201292872645118457648, 12.47797165373275826763555354784, 13.21698876682846169645769145733, 13.43134859164327873650710415583, 14.71321908736066070529856344405, 15.27556558828472035115282221968, 15.8761228954923363840705834725, 16.576427906355561632352454928272, 17.365819775160170611846692472495, 18.116296391900686346760998746255, 18.725889249790206212197122962005, 19.81749420415329768190077727947, 20.53879166893979193101736951230

Graph of the $Z$-function along the critical line